Last week, there was an important conference/school at LMU in Munich simply entitled “*Mathematical Foundations of Physics”*. A mesmerizing concentrate of the Physics heterodoxy where the deviant becomes momentarily the norm. This event was but an excuse to honor Detlef Dürr, one of most prominent quantum dissidents, who is retiring or, better, “cleverly reallocating intellectual ressources away from teaching and administration”.

It is a difficult task to summarize the content of all the talks, but I think it is worthwhile to give an account of roughly what was discussed. The main focus was on alternative approaches both to Quantum Mechanics and Electrodynamics (classical and quantum) as well as to other mathematical physics problems like many-body dynamics and renormalization group techniques.

On the quantum front, the focus was without surprise on (the badly named) *“realist”* approaches: theories where there is something out there that moves according to laws and where the measurement postulate can in principle be logically derived. This includes the de Broglie-Bohm (or pilote wave theory) which had most of its major proponents attending the conference (the mathematical-physicist trio Dürr-Goldstein-Zanghi and all their students as well as the philosophers Maudlin and Esfeld), and collapse models (that were represented mainly by the Trieste school and defended on stage by Angelo Bassi). These two theories give drastically different accounts of the world but quite interestingly, their proponents work together and many physicists, like *e.g.* Roderich Tumulka, have published influential articles discussing the two approaches. This is quite striking in a field that is shattered in tiny communities that tend to hate each other. It comes from the fact that despite their very different mathematical form, the de Broglie-Bohm theory and collapse models can be formulated as dynamical theories of local beables (or primitive ontology), that is as theories of “stuff” moving with mathematically sharp laws that can in principle be applied to the universe as a whole. This contrasts with other interpretations or theories that require an outside observer like the Copenhagen interpretation or that are not formulated in a fully sharp way like the Many-Worlds interpretation or all the different implementations of the Consistent Histories program. The pilot wave theory and collapse models can be disliked, even their proponents doubt they are close to the final story. They are however perfectly intelligible theories with minimal romanticism where all thought experiments are allowed and where every question can in principle be settled with enough mathematical analysis. This is a sufficiently good reason to make them appealing to a handful of mathematicians, physicists, and philosophers (and more modestly to myself).

Now more specifically, on the dBB front the consensus was now that owing to the work of the Bohmian trio and their students, the non-relativistic status of the theory is settled: one understands how the Born rule emerges from typicality arguments and *“equivariance”* (the fact that if the Born rule is true at some moment in time, it is preserved by the equations of motion) and how the operator formalism for observables can be derived in a convincing way (that is as convincing as the derivation of stat-mech from Newtonian mechanics). I think that this has been misunderstood by some participants as meaning that there was nothing to be found anymore on the foundations of non-relativistic quantum mechanics: this simply means that there is nothing *more* to get from the dBB approach in this specific setting. This suggests that physicists now need to attack problems that are considered hard also by the mainstream and by people who do not care about foundations.

In this spirit, Ward Struyve presented some of his bold inquiries into quantum gravity using dBB. Quantum gravity in the Bohmian picture has a nice feature: there is one space-time evolving dynamically. Because of the guiding equations, even if the wave functional on 3-metrics is static in the Wheeler-DeWitt quantization (the so called “*time problem*“), the metric still evolves, time flows. The metric is also always well defined, it is never in a superposition, space-time is as boring as it is classical General Relativity. Questions like “Are there singularities?” become trivially well defined and require zero philosophy. It should be emphasized that it does not solve any technical problem involved in the mathematical quantization of the theory (say at the wave functional or operator level), but simply provides a rather simple picture of the world once such a procedure is provided (in this respect, Ward presented results using the Wheeler-DeWitt quantization and the *Loop* quantization). This is nevertheless sufficient to motivate some excitement, especially as the philosophical debates around Quantum Gravity become more and more fuzzy and almost esoteric.

On the collapse front, the status is a bit different as the current non-relativistic models make predictions which differ from standard Quantum theory. The focus is thus more on phenomenology than on extensions to special or general Relativity (even though I find the latter very interesting in principle, which was the subject of my tiny contribution as a poster). There is now a large variety of experiments which can probe the parameter diagram of collapse models and although the initial proposal of Ghirardi, Rimini and Weber (GRW) is still neither falsified nor validated, one can hope that the case will be closed in the not so distant future. Even if collapse models end up being disproved, the experimental effort will have been worthwhile, probing the possible limits of the superposition principle (collapse models being just one proposal relying on its breakdown).

As usual in these kinds of conferences, there was a panel discussion about the future of the foundations of Quantum Mechanics. The public and the panel both had a large fraction of realists and the debate was made to favor Bohmians and this certainly turned out to be the case. However, the debate was a bit depressing mostly because the questions and comments by the many opponents were at the zeroth order and concerned problems that had been solved a long time ago and sometimes even in the early nineties.

There are no doubt serious open questions with dBB and collapse models (*e.g.* is a theory that generates foliations covariantly really Lorentz covariant?), but they are almost never discussed in such contexts. The reactions to these theories are indeed almost only epidermic: any professor in a related field suddenly feels like an expert who can dismiss all the result while having essentially no knowledge of the recent developments in the field. Realist approaches have some non-intuitive features especially for people who know only of the Copenhagen interpretation and this has motivated almost 30 years of mathematical and philosophical inquiries. This makes naive criticisms based on “gut feeling” likely to have been addressed at some point in the literature. One example of a counter-intuitive result is that it is possible to have consistent pictures of the world with minimalist ontologies, *e.g*. with only point particles without qualities (like charge, spin or mass), where the dynamical content is entirely in the law (which in quantum mechanics means the state vector). One may not like such pictures, but one cannot deny that they work: as counter-intuitive as such a worldview might seem *a priori*, it can be made consistent with our perception of the world (see *e.g.* arXiv:1510.03719). One can repeatedly shout that this theory is “nonsense” as happened during the panel, but one is undoubtedly more convincing with arguments. My hope is that this sad status of affairs is largely generational and will gradually improve as time passes…

Another big part of the conference dealt with electrodynamics which is one of the other theories Detlef Dürr has been exploring. Michael Kiessling first showed that even classical electrodynamics was a very non trivial theory. The Maxwell equations only make sense for smooth matter densities but are ill-defined for point particles. However if one wants to understand effects coming from the electron self-interaction, one would be interested in allowing for something else than smooth matter densities. One can of course smooth out individual particles by giving them a spatial extension, this then gives the Abraham model that breaks Lorentz invariance. One can also modify the Maxwell field equations to handle singular matter distributions (one example is the so called Born-Infeld approach) and this is such a formulation Michael Kiessling is attempting to make rigorous.

The panel discussion on electrodynamics was much more interesting (from my point of view at least). The participants were interested in knowing what were the most promising routes to define mathematically rigorous interacting relativistic theories (classical but mostly quantum). The consensus is now that Quantum Electrodynamics cannot be made rigorous without a cutoff (it is believed that the theory can only become free once such a cutoff is removed). However, numerical simulations of lattice gauge theories suggest that more general Yang-Mills theories are likely to be well defined (even though their construction is a *Millennium problem*). This gives the researchers essentially two options: go to more complicated quantum field theories hoping that the situation will be mathematically better there, or keeping on attacking only quantum electrodynamics and even classical electrodynamics but with necessarily radically different approaches (like the Wheeler-Feynman theory where there are no field and only direct interactions) that would deal with the cutoff problem more efficiently. People like Jürg Fröhlich, strong of their experience in constructive field theory, advocated against the sin they believe they committed in the past: getting too focused on toy models and failing to attack the real issues of the standard model. Younger mathematicians like Dirk Deckert advocated on the contrary for the necessity of solving the mess in the simplest theories first, arguing that it is intellectual laziness to believe all our *mathematical* problems with existing theories will be solved in future *physical* theories. I find both views convincing and it gives different research routes depending on one’s taste.

One less controversial subject which was discussed and which is very actively researched at LMU concerns the simplification of hard many-body equations. The problem consists in deriving effective non-linear dynamics from the large N limit of linear equations in a mathematically rigorous way. Peter Pickl, Aaron Schaal and Dustin Lazarovici presented situations in which such a derivation can be made rigorously: classical (attractive or repulsive) Coulomb interactions (which gives the so called Newton-Vlasov equations), quantum Coulomb interaction (which gives the Schrödinger-Newton equation as a limit) and classical Maxwell-Lorentz equations (which gives the Maxwell-Vlasov equations).

The Schrödinger-Newton equation has a subtle status as it is sometimes taken as a fundamental equation in foundations. As was emphasized by Aaron however, such an equation makes a precise mathematical sense as a limit of linear equations but *we can hope for nothing more*: the equation becomes *inconsistent* once removed from this specific many-body context.

Another interesting talk was delivered by Jean Bricmont on the renormalization group approach to non-linear partial differential equations. The talk was not about new findings but consisted in a pedagogical introduction to the results obtained by Antti Kupiainen and Jean in the nineties (see *e.g.* arXiv:chao-dyn/9411015). This is clearly something I wish I had known before. In the context of PDEs, it is possible to obtain rigorous results about the scaling behavior of solutions using Wilsonian renormalization. Better, it is sometimes even possible to give a meaning to ill-defined stochastic PDEs using renormalization group techniques. This constructive approach is quite beautiful and was recently used by Antti to make sense of the KPZ equation (see arXiv:1604.08712). The setting of PDEs is great to understand the renormalization group arguments in Euclidean and Lorentzian quantum field theories as it is analytically manageable yet sufficiently non-trivial to yield rather deep insights into what is actually going on.

The conference was concluded with recollections by Shelly Goldstein and Stefan Teufel of the impressive work done by Detlef in the last 30 years. It was perhaps even more impressive to see the influence he had had, with his former students filling a whole room. This is, I think, a good occasion to finally dispel the myth that the LMU group of foundations has been excessively *cultish*. I believe the root of this false impression is to be found in my discussion of what happened during the first panel on quantum foundations. As the same debunked rebuttals are opposed over and over again, people necessarily end up giving stereotyped answers out of lassitude. This might give the feeling that young physicists have been conditioned or indoctrinated when they just want to dismiss the zeroth order objections (which have been dealt with a million times) to move to the real open questions. Essentially, the problem is that the exterior signs of a cult are also shared by people who are simply *right* about complicated questions that others have barely bothered considering. This of course cannot be used to prove anyone *is* right, but it at least shows that accusations of sectarianism should not be overused against Bohmians.

Detlef fought against everyone to make LMU a leading center in foundations. When I got interested in these questions about two years ago, it was one of the only place where I could discuss with open-minded people without being told I had gone crazy. I hope the university will conserve at least part of this originality of thought that is so important in academia.

With Detlef’s new life, a page is probably turning for “realists”, but as this conference showed, the field is clearly alive and well. Many young physicists and mathematicians, less prisoners of the old Quantum “Philosophy” than their elders, are now flooding the arena for the best.

AntoineI am glad to inaugurate the comment section of this blog, to which I also would like to subscribe, but I couldn’t find any RSS link!

My comments are probably of the zeroth order, as you know my only familiarity with the subjects discussed here come from our discussions where I’m afraid I was really stubbornly dumb.

In the paragraph about Ward Struyve, “Quantum gravity” means non-relativistic gravity, if I understand correctly, and so it would be clearer to say that we have one space (3d) evolving dynamically (this evolution then produces one spacetime, right, but IMO the generally accepted terminology is to talk about objects at fixed time, eg. one talks about a particle evolving in time, and not about a world-line evolvong in time). Then when you say that space-time is “as boring as in classical GR”, can’t there be much more allowed spaces because Lorentz invariance would not be automatically there?

I find the discussion about electrodynamics, and in particular the fact that it can not be well-defined, while more general (I guess this means non abelian) Yang-Mills can is very interesting. But then what about a Yang-Mills theory with a Higgs mechanism, that would give QED in the infrared? Is it possible that heavy additional fields make QED well-defined?

Thanks for the very nice account of the conference!

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Antoine TilloyPost authorHi Antoine,

On the Quantum Gravity front, Ward was studying simple FLRW spacetimes but to my understanding the analysis would extend (in principle) to the general case. As the quantization is done in the Hamiltonian picture, the main object you consider is the 3-metric but it does not mean that Lorentz invariance is broken (again in principle, this is just a rewriting in the ADM picture). I am not entirely sure how the problem of foliations, which is inherent to dBB, is dealt with in that case: this may indeed produce weirder space-times. However, and contrary to what I’ve been doing with collapse models (maybe you’re referring to our discussions on this), Ward’s results are not confined at all to the Newtonian limit.

By “as boring as GR”, I just mean that you have only one space-time metric. You don’t have complex Schrödinger cat states of metric + matter which makes interpreting the theory difficult. This means that even if weirder space-time emerge, at least there is just one geometry. This is boring in a positive sense, it is straightforwardly intelligible.

As for QED, I think you’re aligned with Jürg. I’m clearly not an expert on the subject, you clearly know much more and I think what you suggest is what some of the participants had in mind. The issue with this program is of course that even non-abelian Gauge theories are still not mathematically well defined. If there is a way to build something that behaves like QED without the Landau pole with only a small change in, say, the classical Maxwell equations, then maybe it is worth studying before going to complicated theories. I think both views make sense and roughly correspond to different communities (physicists for the first option, mathematicians for the second).

By the way, I’ve now added a RSS link and social network sharing buttons…

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